Characteristic property of the product topology

From Maths
Revision as of 00:29, 3 May 2016 by Alec (Talk | contribs) (Creating skeleton for page.)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Characteristic property of the product topology/Statement


TODO: Caption


Let ((Xα,Jα))αI be an arbitrary family of topological spaces and let (Y,K) be a topological space. Consider (αIXα,J) as a topological space with topology (J) given by the product topology of ((Xα,Jα))αI. Lastly, let f:YαIXα be a map, and for αI define fα:YXα as fα=παf (where πα denotes the αth canonical projection of the product topology) then:
  • f:YαIXα is continuous

if and only if

  • βI[fβ:YXβ is continuous] - in words, each component function is continuous

TODO: Link to diagram



Proof

(Unknown grade)
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).

Notes

References